Finite difference method

Abstract: The finite-difference method in the frequency domain is a powerful tool for analyzing arbitrarily shaped transmission-line discontinuities and junctions. An improved formulation based on Maxwell's equations in integral form is presented. It corresponds to the Helmholtz equation and reduces the numerical efforts in solving the large linear equation system considerably. The method is verified by comparison to previous work on microstrip. >

Abstract: The non-linear oscillations of a one-dimensional axially moving beam with vanishing flexural stiffness and weak non-linearities are analysed. The solution of the initial-boundary value problem for the partial differential equation that describes the motion of the beam when two parameters related to the flexural stiffness and the non-linear terms vanish is expanded into a perturbative double series. Two singular perturbation effects due to the small flexural stiffness and to the weak non-linear terms arise: (i) a boundary layer effect when the flexural stiffness vanishes, (ii) a secular effect. Some tests are performed to compare the “first order” perturbative solution with an approximate solution obtained by a finite difference scheme. The effect of the oscillation amplitude combined with the presence of small bending stiffness and axial transport velocity is investigated enlighting some interesting aspects of axially moving systems. The value of the perturbative series as a computational tool is shown.

Abstract: Recently, the numerical modelling and simulation for anomalous subdiffusion equation (ASDE), which is a type of fractional partial differential equation( FPDE) and has been found with widely applications in modern engineering and sciences, are attracting more and more attentions. The current dominant numerical method for modelling ASDE is Finite Difference Method (FDM), which is based on a pre-defined grid leading to inherited issues or shortcomings. This paper aims to develop an implicit meshless approach based on the radial basis functions (RBF) for numerical simulation of the non-linear ASDE. The discrete system of equations is obtained by using the meshless shape functions and the strong-forms. The stability and convergence of this meshless approach are then discussed and theoretically proven. Several numerical examples with different problem domains are used to validate and investigate accuracy and efficiency of the newly developed meshless formulation. The results obtained by the meshless formulations are also compared with those obtained by FDM in terms of their accuracy and efficiency. It is concluded that the present meshless formulation is very effective for the modeling and simulation of the ASDE. Therefore, the meshless technique should have good potential in development of a robust simulation tool for problems in engineering and science which are governed by the various types of fractional differential equations.

Abstract: Like the finite difference method, the finite volume method gives an approximate value for the derivative of a field at a given point using the values of the field at a few locations neighboring the point. The method uses the divergence theorem, considers a “finite volume” around the point and discretizes the surface bounding the volume. When the finite volumes considered are regular polyhedra, one obtains the expressions corresponding to standard centered finite differences, but the finite volume method is more general than the finite difference method because it may deal directly with irregular grids. It is possible to give a finite volume formulation of the elastodynamic problem, using dual volumes, that correspond, in the regular case, to the staggered grids used in the finite difference method. The scheme thus obtained is more general than the one obtained using finite differences, as the “grids” may be totally unstructured, but at the cost of having, in the general case, only a first-order accuracy. Although the scheme is not consistent, numerical tests suggest that it is stable and convergent. This implementation of a finite volume method does not provide a way for a more general treatment of the boundaries than the conventional finite difference method.